Properties

Label 3630.2.a.bi
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + ( -2 - \beta ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + ( -2 - \beta ) q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + ( 1 - \beta ) q^{13} + ( -2 - \beta ) q^{14} + q^{15} + q^{16} + 4 \beta q^{17} + q^{18} + ( -1 + \beta ) q^{19} - q^{20} + ( 2 + \beta ) q^{21} + 3 \beta q^{23} - q^{24} + q^{25} + ( 1 - \beta ) q^{26} - q^{27} + ( -2 - \beta ) q^{28} + ( 6 - 6 \beta ) q^{29} + q^{30} + ( -6 - 2 \beta ) q^{31} + q^{32} + 4 \beta q^{34} + ( 2 + \beta ) q^{35} + q^{36} + ( 1 - \beta ) q^{37} + ( -1 + \beta ) q^{38} + ( -1 + \beta ) q^{39} - q^{40} + ( -2 + 7 \beta ) q^{41} + ( 2 + \beta ) q^{42} + ( -4 + 2 \beta ) q^{43} - q^{45} + 3 \beta q^{46} + ( -1 - \beta ) q^{47} - q^{48} + ( -2 + 5 \beta ) q^{49} + q^{50} -4 \beta q^{51} + ( 1 - \beta ) q^{52} + ( 2 - 5 \beta ) q^{53} - q^{54} + ( -2 - \beta ) q^{56} + ( 1 - \beta ) q^{57} + ( 6 - 6 \beta ) q^{58} + ( -13 - \beta ) q^{59} + q^{60} + ( -8 - 2 \beta ) q^{61} + ( -6 - 2 \beta ) q^{62} + ( -2 - \beta ) q^{63} + q^{64} + ( -1 + \beta ) q^{65} + ( 8 - 10 \beta ) q^{67} + 4 \beta q^{68} -3 \beta q^{69} + ( 2 + \beta ) q^{70} + ( -10 + 6 \beta ) q^{71} + q^{72} + ( 2 - 8 \beta ) q^{73} + ( 1 - \beta ) q^{74} - q^{75} + ( -1 + \beta ) q^{76} + ( -1 + \beta ) q^{78} + ( -2 + 4 \beta ) q^{79} - q^{80} + q^{81} + ( -2 + 7 \beta ) q^{82} + ( -8 + 8 \beta ) q^{83} + ( 2 + \beta ) q^{84} -4 \beta q^{85} + ( -4 + 2 \beta ) q^{86} + ( -6 + 6 \beta ) q^{87} + ( -10 + 3 \beta ) q^{89} - q^{90} + ( -1 + 2 \beta ) q^{91} + 3 \beta q^{92} + ( 6 + 2 \beta ) q^{93} + ( -1 - \beta ) q^{94} + ( 1 - \beta ) q^{95} - q^{96} + ( 2 + 8 \beta ) q^{97} + ( -2 + 5 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 5q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 5q^{7} + 2q^{8} + 2q^{9} - 2q^{10} - 2q^{12} + q^{13} - 5q^{14} + 2q^{15} + 2q^{16} + 4q^{17} + 2q^{18} - q^{19} - 2q^{20} + 5q^{21} + 3q^{23} - 2q^{24} + 2q^{25} + q^{26} - 2q^{27} - 5q^{28} + 6q^{29} + 2q^{30} - 14q^{31} + 2q^{32} + 4q^{34} + 5q^{35} + 2q^{36} + q^{37} - q^{38} - q^{39} - 2q^{40} + 3q^{41} + 5q^{42} - 6q^{43} - 2q^{45} + 3q^{46} - 3q^{47} - 2q^{48} + q^{49} + 2q^{50} - 4q^{51} + q^{52} - q^{53} - 2q^{54} - 5q^{56} + q^{57} + 6q^{58} - 27q^{59} + 2q^{60} - 18q^{61} - 14q^{62} - 5q^{63} + 2q^{64} - q^{65} + 6q^{67} + 4q^{68} - 3q^{69} + 5q^{70} - 14q^{71} + 2q^{72} - 4q^{73} + q^{74} - 2q^{75} - q^{76} - q^{78} - 2q^{80} + 2q^{81} + 3q^{82} - 8q^{83} + 5q^{84} - 4q^{85} - 6q^{86} - 6q^{87} - 17q^{89} - 2q^{90} + 3q^{92} + 14q^{93} - 3q^{94} + q^{95} - 2q^{96} + 12q^{97} + q^{98} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −3.61803 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.38197 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3630.2.a.bi 2
11.b odd 2 1 3630.2.a.bc 2
11.d odd 10 2 330.2.m.d 4
33.f even 10 2 990.2.n.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.d 4 11.d odd 10 2
990.2.n.a 4 33.f even 10 2
3630.2.a.bc 2 11.b odd 2 1
3630.2.a.bi 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7}^{2} + 5 T_{7} + 5 \)
\( T_{13}^{2} - T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( 5 + 5 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -1 - T + T^{2} \)
$17$ \( -16 - 4 T + T^{2} \)
$19$ \( -1 + T + T^{2} \)
$23$ \( -9 - 3 T + T^{2} \)
$29$ \( -36 - 6 T + T^{2} \)
$31$ \( 44 + 14 T + T^{2} \)
$37$ \( -1 - T + T^{2} \)
$41$ \( -59 - 3 T + T^{2} \)
$43$ \( 4 + 6 T + T^{2} \)
$47$ \( 1 + 3 T + T^{2} \)
$53$ \( -31 + T + T^{2} \)
$59$ \( 181 + 27 T + T^{2} \)
$61$ \( 76 + 18 T + T^{2} \)
$67$ \( -116 - 6 T + T^{2} \)
$71$ \( 4 + 14 T + T^{2} \)
$73$ \( -76 + 4 T + T^{2} \)
$79$ \( -20 + T^{2} \)
$83$ \( -64 + 8 T + T^{2} \)
$89$ \( 61 + 17 T + T^{2} \)
$97$ \( -44 - 12 T + T^{2} \)
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